Lesson 13-3: Determinants & Cramer’s Rule

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Lesson 13-3: Determinants & Cramer’s Rule Objective: Students will: Evaluate determinants and Solve systems using Cramer’s Rule

All square (2 x 2, 3 x 3,…) matrices have a determinant 2 x 2 determinant: Difference of products from diagonals (main diagonal 1st) Notation | | Example 1 Solution:-8

Cramer’s Rule (cont’d) then… so… and if needed

Cramer’s Rule Uses determinants to solve systems System: 3x – 2y = -9 x + 2y = 5 D = determinant of coefficient matrix Dx = determinant of matrix with constants in place of x coefficients Dy = determinant of matrix with constants in place of y coefficients

Example 2 Solve Using Cramer’s Rule 4x + 3y = 1 2x + y = 5 D= Dx= Dy= Solution is (7, -9)

3 x 3 Determinant: D = a1 ● (b2c3- b3c2) - a2 ● (b1c3- b3c1) + a3 ●

Solving 3-variable systems: Determinants will be 3 x 3’s Example 3 3x - y + 2z = 1 x – y + 2z = 3 -2x + 3y + z = 3 D= Dx= Dy= Dz= Solution: ( )